Solve the Rational Equation: Isolate X in 5/(x-8) = 3/4x

Q: When can I use cross-multiplication?

Cross-multiplication works perfectly when you have one fraction equals another fraction, like $ \frac{a}{b} = \frac{c}{d} $. This gives you ad = bc.

Q: What if one denominator has parentheses like (x-8)?

Treat the entire expression in parentheses as one unit. When you cross-multiply, you'll need to distribute any coefficient to all terms inside the parentheses.

Q: How do I check if x = -24/17 is correct?

Substitute back: $ \frac{5}{\frac{-24}{17}-8} = \frac{5}{\frac{-160}{17}} $ and $ \frac{3}{4 \cdot \frac{-24}{17}} = \frac{3}{\frac{-96}{17}} $. Both sides should equal the same value!

Q: What if I get x = 0 or x = 8?

Be careful! These values make denominators zero in the original equation. Always check that your solution doesn't create undefined expressions in the original problem.

Cross-Multiplication with Rational Expressions

Solve for X:

$\frac{5}{x-8}=\frac{3}{4x}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solution

00:03 We want to isolate the unknown X

00:07 Let's multiply by both denominators to eliminate fractions

00:14 Let's properly open parentheses, multiply by each factor

00:20 Let's arrange the equation so that one side has only the unknown X

00:29 Let's isolate the unknown X

00:35 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for X:

$\frac{5}{x-8}=\frac{3}{4x}$

Step-by-step solution

To solve the equation $\frac{5}{x-8} = \frac{3}{4x}$ for the variable $x$ , we will follow these steps:

Step 1: Apply cross-multiplication to the equation. This involves multiplying the numerator of each fraction by the denominator of the other fraction:

5 \cdot 4x = 3 \cdot (x - 8)

Step 2: Simplify both sides of the resulting equation:

20x = 3x - 24

Step 3: Rearrange the equation to isolate terms involving $x$ on one side:

20x - 3x = -24

This simplifies to:

17x = -24

Step 4: Solve for $x$ by dividing both sides of the equation by 17:

x = \frac{-24}{17}

Therefore, the solution to the equation is:

$x = \frac{-24}{17}$

Final Answer

$\frac{-24}{17}$

Key Points to Remember

Essential concepts to master this topic

Cross-Multiplication: Multiply numerator of each fraction by opposite denominator
Technique: Transform $\frac{5}{x-8} = \frac{3}{4x}$ to $5 \cdot 4x = 3(x-8)$
Verification: Substitute $x = \frac{-24}{17}$ back into original equation to confirm equality ✓

Common Mistakes

Avoid these frequent errors

Incorrectly distributing during cross-multiplication
Don't write 5 · 4x = 3 · x - 8 = 20x = 3x - 8! This forgets to distribute the 3 to both terms in (x - 8). Always distribute completely: 3(x - 8) = 3x - 24.

Practice Quiz

Test your knowledge with interactive questions

Solve for X:

$$ 3x=18 $$

FAQ

Everything you need to know about this question

When can I use cross-multiplication?

Cross-multiplication works perfectly when you have one fraction equals another fraction, like $\frac{a}{b} = \frac{c}{d}$ . This gives you ad = bc.

What if one denominator has parentheses like (x-8)?

Treat the entire expression in parentheses as one unit. When you cross-multiply, you'll need to distribute any coefficient to all terms inside the parentheses.

Why is my answer negative?

Negative answers are completely normal in rational equations! The key is careful arithmetic - double-check your signs when distributing and combining like terms.

How do I check if x = -24/17 is correct?

Substitute back: $\frac{5}{\frac{-24}{17}-8} = \frac{5}{\frac{-160}{17}}$ and $\frac{3}{4 \cdot \frac{-24}{17}} = \frac{3}{\frac{-96}{17}}$ . Both sides should equal the same value!

What if I get x = 0 or x = 8?

Be careful! These values make denominators zero in the original equation. Always check that your solution doesn't create undefined expressions in the original problem.

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